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What does it imply if F and G are both true?
F
F:& m∠ 1-m∠ 2+m∠ 3=90 G:& m∠ 1+m∠ 2+m∠ 3=180 We can see that m∠ 1+m∠ 3 appear in both of these equation. If we subtract the first equation from the second, both m∠ 1 and m∠ 3 are eliminated. 2m∠ 2=90 ⟺ m∠ 2=45 Since the measure of ∠ 2 is not fixed, this means that the conclusions in F and G do not have to both be true. Let's check the conclusion in G.
G:& m∠ 1+m∠ 2+m∠ 3=180 ∠ 1, ∠ 2, and ∠ 3 are adjacent angles, and together they form a straight angle. The Angle Addition Postulate guarantees that the sum of their angles is the measure of the straight angle, which is 180. Thus, the conclusion in G is always true.
By now we know that:
Notice that in the solution above we did not use the assumption about the congruence of ∠ 1 and ∠ 3. This assumption is needed for showing that the conclusions in H and J are always true. These can be proved using the definition of congruence.
